Number of roots of tanh(ax) = x

[shakgoku]

Homework Statement

what happens to the number of solutions of the equation
$$x = \tanh(\beta x)$$

When
$$\beta$$ is varied from $$\frac{1}{2}$$ to $$\frac{3}{2}$$
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a) unchanged
b) increase by 1
c) increase by 2
d) increase by 3

Homework Equations

$$\tanh(ax) =.... -\frac{17}{315} \, a^{7} x^{7} + \frac{2}{15} \, a^{5} x^{5} - \frac{1}{3} \, a^{3} x^{3} + a x$$


$$tanh(ax)= \frac{e^{ax}-e^{-ax}}{e^{ax}+e^{-ax}}$$

The Attempt at a Solution

Tried to Apply the two above formulas without any success. This is an examination question and too bad, Graphing and programming calculators are not allowed :(

 

[tiny-tim]

hi shakgoku!

what's the shape of y = tanhßx?

what's the shape of y = x?

 

[shakgoku]

hi tiny-tim,

The shape of tanh(βx) is nearly straight line near origin and has horizontal asymptotes at y = 1 and y = -1
The shape of x is straight line through origin , slope = 45 degrees.

so x = 0 is definitely a root. how to find it there are more roots?

 

[tiny-tim]

hi shakgoku!

(just got up :zzz: …)

The shape of tanh(βx) is nearly straight line near origin and has horizontal asymptotes at y = 1 and y = -1

yes, but which ways does it bend?

 

[SammyS]

What is the slope of the line tangent to tanh(ßx) at the origin?

 

[shakgoku]

What is the slope of the line tangent to tanh(ßx) at the origin?

yes, but which ways does it bend?

Thanks , its so obvious now. The slope is at x = 0 is beta. So, if beta<1 the slope keeps decreasing and never gets a chance to intersect y = x again.

But beta > 1, tanh(beta*x) stays > (y = x) but due to decreasing slope, intersects y = x again at two places x = + and - r
(if r is a root).


I've also found out that 0 < r < 1

by taking f = tanh(beta*x) - x and checking its value for 0 and 1. for one value its +ve and for other it was -ve.

 

[tiny-tim]

Thanks , its so obvious now. The slope is at x = 0 is beta. So, if beta<1 the slope keeps decreasing and never gets a chance to intersect y = x again.

But beta > 1, tanh(beta*x) stays > (y = x) but due to decreasing slope, intersects y = x again at two places x = + and - r

yup!

just sometimes we can prove something that looks complicated with hardly any maths!

 

[physicsblr]

hi!
I didnt get that beta>1 part. You mean to say, the equation has 3 roots-0,1,-1? Please let me know, am too stuck with the same problem.

 

[tiny-tim]

draw the graph!

what does it look like?​